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Augmented AMG‐shifted Laplacian preconditioners for indefinite Helmholtz problems
Author(s) -
Tsuji P.,
Tuminaro R.
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1997
Subject(s) - multigrid method , laplace operator , mathematics , helmholtz free energy , solver , linear subspace , helmholtz equation , operator (biology) , convergence (economics)
Summary Discrete representations of the Helmholtz operator generally give rise to extremely difficult linear systems from an iterative solver perspective. This is due in part to the large oscillatory near null space of the linear system. Typical iterative methods do not effectively reduce error components in the subspace associated with this near null space. Traditional coarse grids used within multilevel solvers also cannot capture these components because of their oscillatory nature. While the shifted Laplacian is a popular method allowing multigrid preconditioners to achieve improved convergence, it is still unsatisfactory for higher‐frequency problems. This paper analyzes the shifted Laplacian through polynomial smoothers to show its effect on the spectrum of the discrete Helmholtz operator. This analysis reveals desirable features of the shifted Laplacian preconditioners as well as limitations. Shifted Laplacian techniques are first extended to smoothed aggregation algebraic multigrid. Motivated by analysis, we propose augmenting the algebraic multigrid shifted Laplacian with a two‐grid error correction step. This additional step consists of the generalized minimal residual iterations applied to a projected version of the unshifted equations on a single auxiliary coarse grid. Results indicate that augmentation can improve the convergence significantly and can reduce the overall solve time on two‐dimensional and three‐dimensional problems. Copyright © 2015 John Wiley & Sons, Ltd.